On the other hand, I would agree with you--provided that temperature is indeed continuous wrt time. However,this still leaves the problem of whether the temperature is exactly 80.5 for any finite amount of time. In order for this to be possible, clearly some nth derivative of the function is not continuous.
Now we're starting down the road toward deep metaphysics. For instance:
1. If an event has a duration of zero, did it actually happen? What is the meaning of an instataneous measurement?
2. Since we cannot measure temperatures with ultimate percision, does it make sense at all to say that a temperature is exactly any number? Should a continuous random variable always be confined to a range?
3. Since temperature is a macroscopic phenomenon, does it make sense to apply ultimate percision to it at all? Would the concept of temperature disappear under infinitesmal scrutiny like length does? (See "How Long is the Coastline of Britian?", Benoit Mandelbrot.)
It was never my intention to launch a philosophical discussion. I was just trying to point out that the statement "Zero proability events can occur." is not idiotic. I can easily demonstrate such a case in mathematics, but whether or not you believe it happens in reality is a matter for your personal metaphysics.
"P.S. I do not believe myself to be an idiot. (However, i do not have a proof of this.)
-- Mike"
I believe myself to be an idiot. (I just need to find one counterexample to disprove this. Sounds easier than your way.)
--Chris
...and this of course get us into deep epistimology, the arguments hinging mostly on the definition of "idiot" and the possibility that a true idiot could convince himself that an invalid proof of non-idiotness was valid.
If I recall correctly, the integral approaches zero as the number of steps approaches infinity. Usualy, that's just a technicality, and saying "the integral is" is the same as saying "the integral approaches". But when things get weird, the picky details get significant.
No, no. This integral is exactly zero. You may be thinking of the definition of the Riemann Integral. The Riemann Sums approach the value of the integral as the number of data points approaches infinity.
Integrals, however, do not approach values. The Riemann Integral (which is the one you used in Cal I) explicitly *contains* a limit as n --> infinity. The value of an integral is either a fixed constant, or else the integral diverges (The limit either exists, or it does not).
Remember also that the value of an integral is equal to the area under the curve. An integral with limits the same should be equal to the area of a region with width zero, that is 0. (Here we assume the integral exists and therefore the curve has no singularities.)
"Basic lesson in probability here. If an event has probability zero, it will never occur. Ever. If anyone ever told you an event had probability zero and it did occur, they were an idiot."
Events of probability zero can occur. In fact some of them happened today.
Proof:
Checking the weather data for my city, i see that at 3 PM, the temperature was 82 F, and at 5 PM, it was 79 F. These are not exact values, so let us assume that the temperature was between 81 and 83 at 3 PM and between 78 and 80 at 5 PM.
By the Intermediate Value Theorem, we see that the temperature must have been exactly 80.5 at some time between 3 and 5 PM. (Assuming that temperature behaves continuously, of course.)
Now, assuming that f is the probability density function for temperature in my area, the probability that the temperature is exactly 80.5 at any given time is
P(80.5 = T = 80.5) = int{80.5, 80.5} f(x) dx [1]
Since the limits are the same, the integral is zero (by FTC, re-take Calculus I for details).
The temperature *was* exactly 80.5 at some time today, yet the probability of it being so was zero.
Thus, event of probability zero can happen.
(This is hardly a rigorous proof. I have assumed several things, including the assumption that temperature is not quantized, but i hope the point is taken.)
-- Mike
P.S. I do not believe myself to be an idiot. (However, i do not have a proof of this.)
-- Mike
[1] p 131 _Probability and Statistics for Engineering and the Sciences_, Jay L. Devoe, Brooke/Cole 1991.
"So what you're saying is, there are two kinds of people in the world,those who believe there are two kinds of people in the world and those who don't. Oh Wait.:-) "
The truth is, that's almost exactly what i'm saying. The statement of the theory causes it to transcend logic. We get the same kind of result if you attempt to prove to me that God does not exist. You present a proof, and i might say, "Well, it looks like a correct proof, but since God is omnipotent, He might have adjusted our minds to be fooled by just such an invalid proof." Or i might claim to have psychic powers, but (as is often claimed) they don't work if you try to test them.
None of these things are proofs of existance, of course. They are only proofs that existance cannot be disproved. (For the mathematically inclined: Check out this feature of the Axiom of Choice in ZF.)
My point here is that there is little point in logically debating the existance of a group of people who by definition do not accept logic. This is the sort of thing that one either accepts or rejects based on intuition and experience (or superstition and prejudice as the case may be).
I was introduced to Alan Carter's writings (and his concept of mappers and packers) about a year ago by a mutual acquaintance who independently discovered some of the same concepts, not by working with programmers, but by working with ADD diagnosed children. Since then, i have had numerous discussions about this topic and several points usually come up which i thought i would put here:
1. Alan is not simply labeling two categories of people. He believes that *all* people are born with "mapping" abilities and that these abilities are crushed out of children at an early age. He sees "packing" as a correctable condition, and the Programmer's Stone is intended to do just that (for programmers anyway). He says he has achieved remarkable successes, but i, of course, have no idea.
2. It is extremely difficult to talk about this subject logically. One direct consequence of the description of packers is that they cannot understand mapping. Thus, if you and i disagree, i can always claim that you are a packer and just don't get it. This means that the existance of these two different mindsets can't really be proved. However, it clicks with some people and seems very, very *true* to them.
3. Alan does talk about mapper and packers as if they are discrete, exclusive states, but a lot of people i've talked to believe that people exist along a continuum between these extremes. I'm fairly certain that Alan thinks that as well, but he doesn't explictly say so.
Some of my thoughts here are shaped by things i've read in some of Alan's other writings, especially his "Reciprocality Paper". I am not going to post a link to that paper because the site i got it from seems to disappear periodically, and currently seems to be gone. Also that paper contains an odd Cosmology which i find interesting but which i basically think is a load of nonsense.
Slashdot Mirror
On the other hand, I would agree with you--provided that temperature is indeed continuous wrt time. However,this still leaves the problem of whether the temperature is exactly 80.5 for any finite amount of time. In order for this to be possible, clearly some nth derivative of the function is not continuous.
Now we're starting down the road toward deep metaphysics. For instance:
1. If an event has a duration of zero, did it actually happen? What is the meaning of an instataneous measurement?
2. Since we cannot measure temperatures with ultimate percision, does it make sense at all to say that a temperature is exactly any number? Should a continuous random variable always be confined to a range?
3. Since temperature is a macroscopic phenomenon, does it make sense to apply ultimate percision to it at all? Would the concept of temperature disappear under infinitesmal scrutiny like length does? (See "How Long is the Coastline of Britian?", Benoit Mandelbrot.)
It was never my intention to launch a philosophical discussion. I was just trying to point out that the statement "Zero proability events can occur." is not idiotic. I can easily demonstrate such a case in mathematics, but whether or not you believe it happens in reality is a matter for your personal metaphysics.
"P.S. I do not believe myself to be an idiot. (However, i do not have a proof of this.)
-- Mike"
I believe myself to be an idiot. (I just need to find one counterexample to disprove this. Sounds easier than your way.)
--Chris
...and this of course get us into deep epistimology, the arguments hinging mostly on the definition of "idiot" and the possibility that a true idiot could convince himself that an invalid proof of non-idiotness was valid.
...but that's enough philosophy for today.
-- Mike
If I recall correctly, the integral approaches zero as the number of steps approaches infinity. Usualy, that's just a technicality, and saying "the integral is" is the same as saying "the integral approaches". But when things get weird, the picky details get significant.
No, no. This integral is exactly zero.
You may be thinking of the definition of the Riemann Integral. The Riemann Sums approach the value of the integral as the number of data points approaches infinity.
Integrals, however, do not approach values. The Riemann Integral (which is the one you used in Cal I) explicitly *contains* a limit as n --> infinity. The value of an integral is either a fixed constant, or else the integral diverges (The limit either exists, or it does not).
Remember also that the value of an integral is equal to the area under the curve. An integral with limits the same should be equal to the area of a region with width zero, that is 0. (Here we assume the integral exists and therefore the curve has no singularities.)
Trust me on this: I'm a mathematician.
-- Mike
"Basic lesson in probability here. If an event has probability zero, it will never occur. Ever. If anyone ever told you an event had probability zero and it did occur, they were an idiot."
Events of probability zero can occur. In fact some of them happened today.
Proof:
Checking the weather data for my city, i see that at 3 PM, the temperature was 82 F, and at 5 PM, it was 79 F. These are not exact values, so let us assume that the temperature was between 81 and 83 at 3 PM and between 78 and 80 at 5 PM.
By the Intermediate Value Theorem, we see that the temperature must have been exactly 80.5 at some time between 3 and 5 PM. (Assuming that temperature behaves continuously, of course.)
Now, assuming that f is the probability density function for temperature in my area, the probability that the temperature is exactly 80.5 at any given time is
P(80.5 = T = 80.5) = int{80.5, 80.5} f(x) dx [1]
Since the limits are the same, the integral is zero (by FTC, re-take Calculus I for details).
The temperature *was* exactly 80.5 at some time today, yet the probability of it being so was zero.
Thus, event of probability zero can happen.
(This is hardly a rigorous proof. I have assumed several things, including the assumption that temperature is not quantized, but i hope the point is taken.)
-- Mike
P.S. I do not believe myself to be an idiot. (However, i do not have a proof of this.)
-- Mike
[1] p 131 _Probability and Statistics for Engineering and the Sciences_, Jay L. Devoe, Brooke/Cole 1991.
"So what you're saying is, there are two kinds of people in the world,those who believe there are two kinds of people in the world and those who don't. Oh Wait. :-) "
The truth is, that's almost exactly what i'm saying. The statement of the theory causes it to transcend logic. We get the same kind of result if you attempt to prove to me that God does not exist. You present a proof, and i might say, "Well, it looks like a correct proof, but since God is omnipotent, He might have adjusted our minds to be fooled by just such an invalid proof."
Or i might claim to have psychic powers, but (as is often claimed) they don't work if you try to test them.
None of these things are proofs of existance, of course. They are only proofs that existance cannot be disproved. (For the mathematically inclined: Check out this feature of the Axiom of Choice in ZF.)
My point here is that there is little point in logically debating the existance of a group of people who by definition do not accept logic. This is the sort of thing that one either accepts or rejects based on intuition and experience (or superstition and prejudice as the case may be).
-- Mike
I was introduced to Alan Carter's writings (and his concept of mappers and packers) about a year ago by a mutual acquaintance who independently discovered some of the same concepts, not by working with programmers, but by working with ADD diagnosed children. Since then, i have had numerous discussions about this topic and several points usually come up which i thought i would put here:
1. Alan is not simply labeling two categories of people. He believes that *all* people are born with "mapping" abilities and that these abilities are crushed out of children at an early age. He sees "packing" as a correctable condition, and the Programmer's Stone is intended to do just that (for programmers anyway). He says he has achieved remarkable successes, but i, of course, have no idea.
2. It is extremely difficult to talk about this subject logically. One direct consequence of the description of packers is that they cannot understand mapping. Thus, if you and i disagree, i can always claim that you are a packer and just don't get it. This means that the existance of these two different mindsets can't really be proved. However, it clicks with some people and seems very, very *true* to them.
3. Alan does talk about mapper and packers as if they are discrete, exclusive states, but a lot of people i've talked to believe that people exist along a continuum between these extremes. I'm fairly certain that Alan thinks that as well, but he doesn't explictly say so.
Some of my thoughts here are shaped by things i've read in some of Alan's other writings, especially his "Reciprocality Paper". I am not going to post a link to that paper because the site i got it from seems to disappear periodically, and currently seems to be gone. Also that paper contains an odd Cosmology which i find interesting but which i basically think is a load of nonsense.
-- Mike